Download 3.2 notes

Bellwork
Set up a system, do not solve.
1. There were twice as many students as
adults at the game. All together, there were
2000 people at the game.
2. D bought pickles and chips at the store. He
had 12 items in his bag, and spent $14. Pickles
are $3.50 and chips are $2.50.
Section 3.2
Solving Systems of Equations
Algebraically
Two Methods
1.
2.
Substitution
Elimination
Steps to Substitution
1. Solve one of the equations for one
variable. (Try to solve for the variable
with a coefficient of one.)
2. Substitute the expression from Step 1
into the other equation and solve this
new equation.
3. Substitute the value into one of your
equations to complete the ordered pair.
Examples
1. 3x + 4y = -4
x + 2y = 2
Step 1: Solve the second
equation for x.
Step 2: Substitute this expression
into the other equation and solve.
3x + 4y = -4
3(-2y + 2) + 4y = -4
-6y + 6 + 4y = -4
-2y + 6 = -4
x + 2y = 2
-2y = -10
x = -2y + 2
y=5
Step 3: Complete the ordered pair.
Solution: (-8, 5)
X = -2y + 2
x = -2(5) + 2
x = -8
Examples
2. 3x – y = 13
2x + 2y = -10
Step 1: Solve the first
equation for y.
Step 2: Substitute this expression
into the other equation and solve.
2X + 2y = -10
2x + 2(3x – 13) = -10
2x + 6x – 26 = -10
8x – 26 = -10
3x – y = 13
8x = 16
– y = – 3x + 13
x=2
y = 3x – 13
Step 3: Complete the ordered pair.
Solution: (2, -7)
y = 3x – 13
y = 3(2) – 13
y = -7
Examples
3. 2x + 2y = 12
y = 3x – 5
Step 1: The second equation
is already solved for y.
Step 2: Substitute this expression
into the other equation and solve.
2X + 2y = 12
2x + 2(3x – 5) = 12
2x + 6x – 10 = 12
8x – 10 = 12
y = 3x – 5
8x = 22
x = 11/4
Solution: (
/4, 13/4)
11
Step 3: Complete the ordered pair.
y = 3x – 5
y = 3(11/4) – 5
y = 13/4
Examples
4. - 4x – 2y = 16
x + ½ y = -3
Step 1: Solve the second
equation for x.
x + ½ y = -3
x = -1/2 y – 3
No Solution
Step 2: Substitute this expression
into the other equation and solve.
-4x – 2y = 16
– 4(- ½ y – 3) – 2y = 16
2y + 12 – 2y = 16
0 + 12 = 16
0 =4
This is not a true statement and our
variables canceled, so there is no
solution.
Steps to Elimination
1. Make one of the variables have opposite
coefficients by multiplying by a constant.
2. Add the equations together and solve for the
remaining variable.
3. Substitute the value into one of your
equations to complete the ordered pair.
Examples
1. 2x – 4y = 13
4x – 5y = 8
Step 1: Make one of the
variables opposites.
Step 2: Add the equations together
and solve.
-4x + 8y = – 26
+
4x – 5y = 8
3y = -18
y = -6
(2x – 4y = 13) ● - 2
(4x – 5y = 8)● 1
Step 3: Complete the ordered pair.
2X – 4y = 13
2x – 4(-6) = 13
Solution:
(- 11/2, -6)
2x + 24 = 13
2x = -11
x = -11/2
Examples
2. x + 2y = – 2
3x – 2y = 10
Step 1: Make one of the
variables opposites.
The y's are already
opposites.
Step 2: Add the equations together
and solve.
X + 2y = – 2
+
3x – 2y = 10
4x
x= 2
Step 3: Complete the ordered pair.
X + 2y = – 2
2 + 2y = – 2
Solution:
(2, -2)
= 8
2y = -4
y = -2
Examples
3. 2x + 3y = 2
3x – 4y = -14
Step 1: Make one of the
variables opposites.
Step 2: Add the equations together
and solve.
-6x – 9y = – 6
+
6x – 8y = – 28
– 17y = -34
(2x + 3y = 2) ● - 3
y=2
(3x – 4y = -14)● 2
Step 3: Complete the ordered pair.
2X + 3y = 2
2x + 3(2) = 2
Solution:
(-2, 2)
2x + 6 = 2
2x = -4
x = -2
Examples
4. 3x + 2y = 2
6x + 4y = 4
Step 1: Make one of the
variables opposites.
(3x + 2y = 2) ● - 2
(6x + 4y = 4)● 1
Solution:
IMS
Step 2: Add the equations together
and solve.
-6x – 4y = – 4
+ 6x + 4y = 4
0=0
All the variables canceled and its a
true statement so these are the
same lines and the solution is
infinitely many solutions.
Word Problems
1. Your family is planning a 7 day trip to Florida.
You estimate that it will cost $275 per day in
Tampa and $400 per day in Orlando. Your total
budget for the 7 day is $2300. How many days
should you spend in each location?
X = # of days in Tampa
Y = # of days in Orlando
X+y=7
275x + 400 y = 2300
Word Problems
2. You plan to work 200 hours this summer
mowing lawns or babysitting. You need to make a
total of $1300. Babysitting pays $6 per hour and
lawn mowing pays $8 per hour. How many hours
should you work at each job?
X = # of hours babysitting
Y = # of hours of mowing
X + y = 200
6x + 8y = 1300
Word Problems
3. You make small wreaths and large wreaths to
sell at a craft fair. Small wreaths sell for $8 and
large wreaths sell for $12. You think you can sell
40 wreaths all together and want to make $400.
How many of each type of wreath should you
bring to the fair?
X = # small wreaths
Y = # large wreaths
X + y = 40
8x + 12y = 400
Homework Answers
11. (4, -1)
27. IMS
12. No Sol
29. (1/3, 1)
13. (3,3)
31. (18/41, 605/82)
14. (22/5, -6)
33. No Sol
15. (0, 5/2)
35. (-5, -2)
17. (-12/5, 51/5)
37. (5,0)
22. IMS
39. No sol
23. (-11/3, -1)
41. (-69/11, 65/11)
25. (0, 4/5)
43. (-25/4, 5/2)